At this point, the only examples we have of groups that contain expanders, or more generally sequences of graphs of large girth, are constructed using the methods of small cancellation - specifically they come from different small cancellation labellings of sequences of finite graphs that have large girth. This was accomplished first by Gromov [1]}, which was discussed, detailed and expanded by Arzhantseva–Delzant [2]}. The later examples, which also rely on small cancellation (although of a different flavour), were shown to contain examples with the Haagerup property - this is due to work of Arzhantseva–Osajda [3]} and Osajda [4]}.
